Mid-height deflection of a bar - Euler's Formula

In summary, the conversation discusses a question about a straight, vertical aluminum bar that is axially loaded until it buckles. The question asks for the mid-height deflection of the bar before it reaches its plastic yield stress. The conversation includes relevant equations and hints to help find the solution. It also discusses the ambiguity of the question and potential interpretations.
  • #1
SaRaH...
7
0

Homework Statement


I think I've got part of this question but it's multiple choice and nothing that I've got matches any of the options we were given. I'd really appreciate it if you could help me out.

A straight, vertical aluminium bar, 1.0-m in length and 12.5-mm x 4.8-mm in cross
section, is axially loaded until it buckles. Assuming Euler’s formula applies, determine
the mid-height deflection, in millimetres, of the vertical bar before the material attains
its plastic yield stress of 250-MPa.

Homework Equations



P = EIpi2/L2

I = bd3/12

deflection = PL/AE

The Attempt at a Solution



I = (12.5)(4.8)3/12 = 115.2mm4

P = (70*109)(115.2*10-12)*pi2/(1)2 = 79.6N

Then when I tried to get deflection it came out as a huge answer. I'm not certain if that's the right formula I'm using but it's the only one we've used in class so I don't know what else it could be.

The answers we were given were 57mm, 150mm, 31mm, 145mm, 378mm

Sarah
 
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  • #2
SaRaH...: Hint 1: Your third relevant equation is inapplicable; you instead need the bending stress formula. Hint 2: Bending moment is force times distance. You correctly computed the force.

By the way, always leave a space between a numeric value and its following unit symbol. E.g., 79.59 N, not 79.59N. And, e.g., 250 MPa, not 250-MPa. See the international standard for writing units (ISO 31-0).
 
  • #3
For what it's worth, I found that the section is FULLY plastic when deflection is 45 mm. So you could check whether extreme fibres first reach yield stress at 31 mm. I think the question is a bit ambiguous, but it's a well intentioned question.
 
  • #4
pongo38: I am thinking your statement currently appears incorrect, unless I am misinterpreting. Would you be able to show how you obtained your answer? I currently did not find the question ambiguous.
 
  • #5
Is th question asking for the deflection when the extreme fibres FIRST reach their yield stress, or when the whole section has gone plastic (about 50% more moment required)?
 
  • #6
pongo38: Aluminum has no yield plateau. I interpreted the question as meaning it is asking for the deflection when only the extreme fibre reaches the tensile yield strength, Sty = 250 MPa.
 
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What is mid-height deflection of a bar?

Mid-height deflection of a bar refers to the amount by which a bar or beam bends or deflects at its midpoint when a load is applied.

What is Euler's formula?

Euler's formula, also known as the Euler-Bernoulli beam theory, is a mathematical equation used to calculate the deflection of a beam or bar under a load. It states that the mid-height deflection of a bar is directly proportional to the applied load, the length of the bar, and the fourth power of the bar's cross-sectional area, and inversely proportional to the elastic modulus of the material.

What is the significance of mid-height deflection in structural design?

The mid-height deflection of a bar is important in structural design as it helps engineers determine the maximum allowable load a structure can withstand without experiencing excessive deflection, which can lead to failure. It also allows for the selection of appropriate materials and dimensions for a given application.

What are the assumptions made in Euler's formula?

Euler's formula is based on several assumptions, including the bar or beam being made of a homogeneous and isotropic material, having a constant cross-sectional area, and being subjected to a load that does not cause significant bending in the longitudinal direction. It also assumes that the deflection is small compared to the length of the bar, and the load is applied slowly and gradually.

How accurate is Euler's formula?

Euler's formula provides a good estimate of the mid-height deflection of a bar under a load, but it is not completely accurate. It does not account for factors such as shear deformation, lateral buckling, and material nonlinearity, which can affect the actual deflection of a bar. Therefore, it should be used as a preliminary design tool and verified with more accurate methods such as finite element analysis or experimental testing.

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